#8429523

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,7)(3,4)(11,16)(12,15)(19,24)(20,23)(26,31)(28,34)(32,38)(36,40);(1,2)(3,5)(4,8)(6,7)(9,10,14,18)(11,12,16,15)(13,17,22,21)(19,20,24,23)(27,35)(32,38)(36,40)(37,39);(1,3,6,4)(2,5,7,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,6,4)(2,8,7,5)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(1,4,6,3)(2,8,7,5)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,5,6,8)(2,3,7,4)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336719);